Page 757 - Mechanical Engineers' Handbook (Volume 2)

748 State-Space Methods for Dynamic Systems Analysis

the system is completely state controllable. The transformation matrix T in Tables 2 and 4
required to transform the state-space equations for a SISO system to the controllable canon-
ical form exists if and only if the n n matrix P is nonsingular; that is, the system is
c
controllable. Equivalent controllability conditions that are simpler to evaluate than the one
previously stated are also listed in Table 8 along with a necessary (but not sufficient) con-
dition for complete controllability.
The controllability conditions for LTV systems over a specified time interval are more
cumbersome to evaluate in practice as they involve the system transition matrix. These
4
conditions are listed in Table 8. In contrast to time-invariant systems, the controllability of
time-varying systems depends on the time interval under consideration.
The concept of output controllability, as opposed to state controllability described earlier,
was introduced by Kreindler and Sarachik. 13 A linear, continuous-time system is said to be
output controllable at time t if there exists a finite time t t and a control function u(t),
1
0
0
t t t , that drives the system output from any initial value y(t )toany final value y(t ).
0
0
1
1
If this condition holds true for all times t , the system is completely output controllable.
0
Extension of the concept to linear, discrete-time systems is straightforward as before.
4
Output controllability conditions for linear systems that are purely dynamic [i.e., the
matrix D 0 in Eqs. (5), (7), (11), and (13)] are summarized in Table 8. These conditions
are weaker than the corresponding conditions for state controllability if the number of outputs
p is less than the number of state variables n. Since this is true in practice, state controllability
implies output controllability. On the other hand, output controllability does not imply state
controllability in general. It can be shown, however, that for time-invariant systems if the
matrix (CC ) is nonsingular, output controllability is equivalent to state controllability.
T
The observability of a linear system is a measure of the coupling between the system
11
state and its outputs. The concept of observability was introduced by Kalman and is relevant
to the problem of estimation of system state based on the output vector. The output vector
is usually chosen to correspond to measurable variables.
A linear, continuous-time system is said to be observable at time t if there exists a
4
0
finite time t t such that x(t ) can be determined from the history of inputs u(t) and
1
0
0
outputs y(t) over the time interval t t t . If the system is observable for all times t 0
1
0
and all initial states x(t ), the system is completely observable. Extension of the observability
0
concept to discrete-time systems simply requires that the sequence numbers k, k , k be
1
0
substituted for the times t, t , t , respectively, in the previous definitions. A stronger form of
0
1
observability for LTV systems is that of uniformly complete observability. The mathematical
definition of this form of observability is given by Kalman. 11 This property guarantees that
the time interval required to estimate the state is relatively independent of the initial time.
For LTI systems, of course, complete observability is the same as uniformly complete ob-
servability. A property complementary to observability for LTV systems is that of reconstruc-
tibility, 12,14 which concerns the estimation of the state of the system from past measurements
of the state. In contrast to this, observability concerns the estimation of the state from future
measurements of the output. For time-invariant systems, the two properties of reconstructi-
bility and observability are identical to one another.
As was the case for controllability, direct application of the definition of observability
4
already stated yields conditions involving the transition matrix. Simpler algebraic conditions
are available for time-invariant systems. The observability condition for LTI systems with
distinct eigenvalues can be stated very simply if the state equations are transformed to the
Jordan canonical form. Such systems are completely observable if each column in the trans-
4
formed C matrix has at least one nonzero element. The presence of a column of zeros in
this matrix would indicate that the corresponding state variable cannot be estimated from
the measured output and input vectors.